#6516830

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500120
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(13,14,18,24,19,22,17,23)(15,25,29,20,27,36,28,35)(16,30,32,37,33,38,31,21)(26,40,43,44,41,34,42,39);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,29)(18,28)(19,27)(21,34)(22,35)(23,36)(24,25)(30,44)(31,43)(32,42)(33,41)(37,40)(38,39);(6,9)(7,11)(13,16,19,33)(14,21,22,37)(15,26,27,41)(17,32,18,31)(20,34,35,40)(23,30,24,38)(25,39,36,44)(28,43,29,42);(5,7,8,11)(6,10,9,12)(14,22)(15,27)(21,37)(23,24)(26,41)(28,29)(30,38)(42,43);(6,9)(10,12)(14,22)(20,35)(21,37)(23,24)(25,36)(30,38)(34,40)(39,44)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6424026);